Sizes of witnesses in Covtree
Jette Gutzeit, Kimia Shaban, Karen Yeats, and Stav Zalel

TL;DR
This paper investigates the size bounds of minimal witnesses in poset structures related to quantum gravity, introducing the exchange graph as a new analytical tool and establishing specific size bounds.
Contribution
It introduces the exchange graph of downsets and proves new bounds on the size of minimal witnesses, including a non-linear upper bound and specific cases for k=3.
Findings
No linear upper bound of the form n+k+c exists for minimal witnesses.
All minimal witnesses satisfy |Q| ≤ nk - n.
For k=3, there exists a minimal witness with size ≤ 1.5(n+1).
Abstract
Given a set of unlabelled posets, each of size , we say that a poset is a \emph{witness} to if is the set of downsets of size of . We say that is a \emph{minimal witness} if it does not contain a proper downset that is itself a witness to . Motivated by the causal set approach to quantum gravity, we study the upper bound on the size of minimal witnesses as a function of and . We show that there is no linear upper bound of the form for any constant . We introduce the \emph{exchange graph of downsets} as a new tool to study this scenario, and use it to show that all minimal witnesses satisfy the bound , and that when there is at least one minimal witness that satisfies the bound .
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